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Author Archives: jgl
Zhao, Xi and Chen’s wellfiltered, nonsober dcpo
There are several known examples of dcpos that are wellfiltered, but not sober, and I have already mentioned one due to Xiaodong Jia. I would like to explain another one, due to Dongsheng Zhao, Xiaoyong Xi, and Yixiang Chen. This … Continue reading
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QuasiPolish spaces as rounded ideal completions
This month, a pearl by Matthew de Brecht. It is known that the rounded ideal completion of an abstract basis (a set B with a transitive, interpolative relation) is a continuous dcpo, and that all continuous dcpos can be obtained … Continue reading
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Convergence without points
Can you define convergence without mentioning points? More precisely, is there any form of Stone duality for convergence spaces, instead of just topological spaces? The short answer is yes. For the long answer, read the full post.
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X. Jia’s wellfiltered, nonsober dcpo
[Business as usual, despite all viruses!] Peter Johnstone once showed the existence of a dcpo J that is not sober in its Scott topology. That dcpo is not wellfiltered either. Is there a dcpo that is not sober but is … Continue reading
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Dcpos built as graphs of functions
Let X and P be two dcpos, and let ψ be a map from X to P. When is the graph of ψ a dcpo? I will give you a funny sufficient condition, which involves the socalled dtopology, and Hausdorffness. … Continue reading
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Borel sets, analytic sets, and the Baire property
Let me venture into the realm of σalgebras. Yes, you might say, that is measure theory, not topology… but topology plays an important role in measure theory and, for that matter, descriptive set theory. I will tell you about sets … Continue reading
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The RasiowaSikorski lemma and the Baire property
Merry Christmas! And a Happy New Year, too. There are incredible links between logic and topology, and I would like to start with something called the RasiowaSikorski lemma in logic. That is a theorem that states the existence of certain … Continue reading
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Quotients, colimits of dcpos, and related matters
How do we build colimits in the category Top of topological spaces? This is easy: we take the quotient of a big disjoint sum. How do we build colimits in the category Dcpo of dcpos? This is a much more … Continue reading
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Corecompact+wellfiltered T0=sober locally compact
Last time, I motivated the construction of the wellfilterification Wf(X) of a space X of X. Xu, Ch. Shen, X. Xi and D. Zhao by saying that it was needed to understand their proof of the fact that every corecompact … Continue reading
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Wellfilterifications
Xiaodong Jia once asked the following question: is every corecompact, wellfiltered space automatically locally compact? The question was solved positively this year by J. Lawson and X. Xi. I originally planned to try and explain their result. Even more recently, … Continue reading
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